Page 215 - Schaum's Outline of Differential Equations

198 SOLVING SECOND-ORDER DIFFERENTIAL EQUATIONS [CHAP. 20

20.5. Use Euler's method to solve /' - y = x; y (0) = 0, /(O) = 1 on the interval [0,1] with h = 0.1.
Using the results of Problem 20.1, we have/(jt, y, z) = z, g(x, y,z)=y + x,x 0 = 0, y 0 = 0, and z 0 = 1. Then, using
(20.3), we compute

Continuing in this manner, we generate Table 20-1.

Table 20-1

Method: EULER'S METHOD

Problem: y" - y = x; y(0) = 0, /(O) = 1
ft = 0.1
x n
True solution
y n Zn Y(x) = e -e~ -x
x
x
0.0 0.0000 1.0000 0.0000
0.1 0.1000 1.0000 0.1003
0.2 0.2000 1.0200 0.2027

0.3 0.3020 1.0600 0.3090
0.4 0.4080 1.1202 0.4215

0.5 0.5200 1.2010 0.5422
0.6 0.6401 1.3030 0.6733

0.7 0.7704 1.4270 0.8172
0.8 0.9131 1.5741 0.9762

0.9 1.0705 1.7454 1.1530
1.0 1.2451 1.9424 1.3504

20.6. Use Euler's method to solve y" - 3y' + 2y = 0; y(0) = -1, /(O) = 0 on the interval [0, 1] with
A = 0.1.
Using the results of Problem 20.2, we have/(jc, y, z) = Z, g(x, y, z) = 3z — 2y, x 0 = 0,y 0 = —1, and